Abadir, Karim MKarim MAbadir2022-09-012023-01The Econometrics Journal, 2023, 26 (1), pp.88-1041368-4221http://hdl.handle.net/10044/1/99338We propose a minimal representation of variance matrices of dimension k, where parameterization and positive-definiteness conditions are both explicit. Then, we apply it to the specification of dynamic multivariate volatility processes. Compared to the most parsimonious unrestricted formulation currently available, the required number of covariance parameters (hence processes) is reduced by about a half, which makes them estimable in full parametric generality if needed. Our conditions are easy to implement: there are only k of them, and they are explicit and univariate. To illustrate, we forecast minimum-variance portfolios and show that risk is always reduced (by a factor of 2 to 3 in spite of us using the simplest dynamics) compared to the standard benchmark used in finance, while also improving returns on the investment. Because of our representation, we do not get the usual dimensionality problems of existing unrestricted models, and the performance relative to the benchmark is actually improved substantially as k increases.© The Author(s) 2022. Published by Oxford University Press on behalf of Royal Economic Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.Journal Articlehttps://www.dx.doi.org/10.1093/ectj/utac023https://academic.oup.com/ectj/advance-article/doi/10.1093/ectj/utac023/66758071368-423X